Meromorphic function

A meromorphic function is holomorphic on an open subset of the complex plane except possibly at isolated poles. At a pole a of order m, the function has a Laurent expansion whose negative-power part ends with c_-m/(z−a)^m, where c_-m is nonzero; all remaining terms form a holomorphic function near a.² The value is often recorded as infinity at the pole, making the function a holomorphic map into the Riemann sphere.

A removable singularity can be filled in by continuity and therefore does not remain an exception after extension. An isolated essential singularity, by contrast, is forbidden: functions such as exp(1/z) are holomorphic away from zero but are not meromorphic there. The reciprocal 1/f is meromorphic wherever a nonzero holomorphic function f has isolated zeros, with the order of each pole equal to the multiplicity of the corresponding zero.

Laurent series provide the local test for meromorphicity, while partial fractions describe many global examples. Every rational function is meromorphic on the complex plane, and its behavior at infinity determines whether it extends meromorphically to the Riemann sphere: polynomials have a pole at infinity unless they are constant, while a proper rational function has a removable value there.

The Mittag-Leffler theorem gives a converse in a broad sense: on suitable domains, one can prescribe isolated poles and principal parts and construct a meromorphic function realizing them, subject to convergence adjustments.² On a compact Riemann surface, however, global meromorphic functions are constrained; on the sphere they are exactly rational functions. A nonconstant meromorphic function on a compact surface is surjective onto the sphere and has finitely many zeros and poles, counted with multiplicity.

Zeros and poles control the most useful calculus of meromorphic functions. The residue at a pole is the coefficient of (z−a)⁻¹ in its Laurent expansion, and contour integrals depend only on these coefficients through the residue theorem.¹ The argument principle converts the change of argument of a meromorphic function around a contour into the number of zeros minus the number of poles inside, with multiplicities.

These facts support contour evaluation of real integrals, asymptotic estimates, and analytic continuation. The gamma function is meromorphic with simple poles at the nonpositive integers, while the Riemann zeta function has a meromorphic continuation with one simple pole at 1.¹ In complex dynamics, rational maps are meromorphic self-maps of the sphere, and their poles participate in the study of iteration and Julia sets.

Meromorphicity is preserved by addition, multiplication, and division unless the denominator is identically zero; cancellation can turn an apparent pole into a removable singularity. A quotient of two holomorphic functions is therefore a standard source of examples, but local cancellation must be checked rather than inferred from a displayed formula.

On the complex plane, a meromorphic function may have infinitely many poles, provided they have no finite accumulation point. The Weierstrass elliptic functions illustrate a stricter geometry: they are doubly periodic meromorphic functions whose poles repeat on a lattice, and every elliptic function has equally balanced total numbers of zeros and poles in a fundamental parallelogram. Meromorphic functions also underlie Nevanlinna theory, which measures how often a function assumes values and how its poles and zeros contribute to its growth.⁴

Pole

An isolated singularity at which the magnitude of a function tends to infinity like a finite power of the reciprocal distance.

Principal part

The negative-power portion of a Laurent expansion around an isolated singularity.

Residue

The coefficient of (z−a)−1 in the Laurent expansion at a.

Mittag-Leffler theorem

A theorem guaranteeing, under convergence conditions, a meromorphic function with prescribed isolated principal parts.

Riemann sphere

The complex plane together with one point at infinity, viewed as a compact Riemann surface.